Comparison.  It was shown in [1]}, [2]} that PACBayes-sharpness-based measures outperform norm-based measures in terms of the Kendall's rank coefficient and distributional robustness, which is consistent with our findings.
It is worth noting that, however, our experiment settings are different from theirs.
In particular, we only change the initialization scheme and keep the other hyperparameters (e.g. batch size, network architecture, learning rate) the same.
This is important in the sense that some of the generalization bounds highly depend on the hyperparameters, but the expressions of the corresponding complexity measures do not reflect such dependencies.
For example, the distance-based generalization bounds in [3]}, [4]}
rely on the depth of the neural networks,
while the distance-based measure (equation (23) in [2]}) is barely the Frobenius distance without a depth-dependent factor.
Another example is the length-based measure (REF ), where the term \(L_{\Psi }\)  depends on the network architecture.
Thus, to make the complexity measure consistent with the generalization bound, we only vary the initialization scheme and do not compare the Kendall's rank coefficient across the models.
We are therefore not proposing (REF ) as a universal generalization gap predictor.
